Question

If we hold 44% in the risky market portfolio, M, and 56% in the risk-free
an asset with a risk-free rate of 2%, the expected return on the market of
10% and the standard deviation of the market is 3%. Find the expected
return on the portfolio ( ERp) and the standard deviation of the portfolio (σp)

the answer is ERp=5.52%, σp= 1.32%. Could you please in solution (How to Solve)????

Answer 1

Weight of a risky market portfolio = w₁ = 44%, expected return on the market = R₁ = 10%, standard deviation of the market = σ₁ = 3%

Weight of risk-free asset = w₂ = 56%, Expected return on the risk-free asset = R₂ = 2%, Standard-deviation of the risk-free asset = σ₂ = 0 [Risk free asset carry zero risk]

Expected Return

Expected return on the overall portfolio is calculated using the formula:

E[R_P] = w₁*R₁ + w₂*R₂ = 44%*10% + 56%*2% = 5.52%

Variance on the overall portfolio is calculated using the formula:

σ_P² = w₁²*σ₁² + w₂*σ₂² + 2*w₁*w₂*ρ*σ₁*σ₂

where ρ is correlation coefficient between market portfolio and risk-free asset

Since σ₂ = 0

We get, σ_P² = w₁²*σ₁² + 0 + 0 = w₁²*σ₁²

Standard Deviation

Standard deviation is the square-root of variance

So, standard deviation of the overall portfolio = σ_P = [w₁²*σ₁²]^1/2 = w₁*σ₁

w₁ = 44%, σ₁ = 3%

Standard deviation of the portfolio = σ_P = w₁*σ₁ = 44%*3% = 0.0132 = 1.32%

Answer

Expected return of the portfolio = E[R_P] = 5.52%

Standard deviation of the portfolio = σ_P = 1.32%